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Another graph-based method for superpixel segmentation was proposed by Lui et al. Using greedy optimization, summarized in algorithm 1, an objective function based on the entropy rate of a random walk on the graph $\hat{G} = (V,M)$ with $M \subseteq E$ is proposed (where we interpret the image $I$ as 4-connected graph $G = (V,E)$):

$E(\hat{G}) = H(\hat{G}) + \lambda B(\hat{G})$

where $H(\hat{G})$ refers to the entropy rate of the randon walk, while $B(\hat{G})$ defines a balancing term. The objective is maximized subject to the constraint that the number of connected components in $\hat{G}$ is equal or lower to the desired number of superpixels $K$. Given weights $w_{n,m}$ between pixels $x_n$ and $x_m$, defined using a Gaussian kernel based on the $L_1$ color distance, $H(\hat{G})$ is defined as:

where $S_i$ denotes the $i^\text{th}$ superpixel. Starting from an initial superpixels segmentation where each pixel forms its own superpixel, the algorithm greedily adds edges to merge superpixels until the desired number of superpixels is reached, see algorithm 1.